 Grand canonical ensemble

The grand canonical ensemble describes equilibrium with a reservoir that fixes temperature $T$ and chemical potential $\mu$, while the system's volume is fixed and both energy and particle number can fluctuate. A microstate with energy $E$ and particle number $N$ has probability proportional to $e^{-\beta (E-\mu N)}$, with $\beta =1/(k_{B}T)$. The normalization is the grand canonical partition function $\Xi$.

A small region of gas exchanging particles and heat with a much larger gas reservoir illustrates the ensemble. Differentiating $\log \Xi$ gives $⟨N⟩=k_{B}T{\partial }_{\mu }\log \Xi$, and a second derivative gives the number variance. Thus a response to chemical potential and spontaneous particle-number fluctuations encode related information. The ensemble does not mean that total particle number in system plus reservoir is unconstrained; fluctuations describe the chosen subsystem.